According to mathworld
41,42. "Derivatives $\zeta^{(n)}(1/2)$ can also be given in closed form"
with example for the first derivative.

What is the closed form? References?

The motivation is that this question
expresses $\zeta(3)$ in terms of $\zeta(1/2)$ and the first 3 derivatives,
so closed form possibly might result in closed form for zeta(3)
(unless the closed form is derived by the linked question).

Actually you only get every other derivative for free this way. The functional equation says $\xi(s) = \pi^{s/2} \Gamma(s/2) \zeta(s)$ is symmetric about $s = 1/2$, so its odd-order derivatives vanish there, which gives linear equations on the $\zeta^{(n)}(1/2)$ that let you solve for $\zeta^{(2m+1)}(1/2)$ as a linear combination of $\zeta(1/2)$, $\zeta''(1/2)$, $\zeta^{(4)}(1/2)$, ..., $\zeta^{(2m)}(1/2)$. But you still can't solve for the even-order derivatives in terms of derivatives of lower order.
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Noam D. ElkiesMay 10 '13 at 19:59